LMFDB - Embedded newform 2960.2.a.x.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2960,2,Mod(1,2960)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2960 = 2^{4} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2960.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$23.6357189983$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.998068.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 10x^{2} + 3x - 2 $ x^5 - 2x^4 - 6x^3 + 10x^2 + 3x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1480)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.26473$ of defining polynomial
Character	$\chi$	$=$	2960.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.612608 q^{3} -1.00000 q^{5} -3.03374 q^{7} -2.62471 q^{9} +O(q^{10})$	$q-0.612608 q^{3} -1.00000 q^{5} -3.03374 q^{7} -2.62471 q^{9} +0.741603 q^{11} +4.52946 q^{13} +0.612608 q^{15} +2.10832 q^{17} +3.77534 q^{19} +1.85849 q^{21} -5.48321 q^{23} +1.00000 q^{25} +3.44575 q^{27} +9.90434 q^{29} +4.71327 q^{31} -0.454312 q^{33} +3.03374 q^{35} -1.00000 q^{37} -2.77478 q^{39} -2.99959 q^{41} -12.1873 q^{43} +2.62471 q^{45} -5.73788 q^{47} +2.20358 q^{49} -1.29158 q^{51} +9.09209 q^{53} -0.741603 q^{55} -2.31281 q^{57} -5.06692 q^{59} -15.0296 q^{61} +7.96269 q^{63} -4.52946 q^{65} -3.55869 q^{67} +3.35906 q^{69} -10.1542 q^{71} -8.15417 q^{73} -0.612608 q^{75} -2.24983 q^{77} +2.00913 q^{79} +5.76324 q^{81} -11.3210 q^{83} -2.10832 q^{85} -6.06748 q^{87} +3.26656 q^{89} -13.7412 q^{91} -2.88739 q^{93} -3.77534 q^{95} +2.18735 q^{97} -1.94649 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} - 5 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) $ 5 * q - q^3 - 5 * q^5 + 2 * q^7 + 6 * q^9	$ 5 q - q^{3} - 5 q^{5} + 2 q^{7} + 6 q^{9} - 8 q^{11} - 4 q^{13} + q^{15} - q^{17} - 10 q^{19} + 5 q^{21} - 4 q^{23} + 5 q^{25} - q^{27} + 11 q^{29} + 3 q^{31} - 3 q^{33} - 2 q^{35} - 5 q^{37} - 18 q^{39} + 16 q^{41} - 31 q^{43} - 6 q^{45} - 5 q^{47} + 7 q^{49} - 34 q^{51} + 8 q^{53} + 8 q^{55} - 8 q^{57} - 24 q^{59} - 15 q^{61} - 19 q^{63} + 4 q^{65} - 12 q^{67} + 10 q^{69} - 5 q^{71} + 5 q^{73} - q^{75} - 4 q^{77} - 4 q^{79} + 17 q^{81} - 27 q^{83} + q^{85} + 4 q^{87} + 16 q^{89} - 26 q^{91} + 14 q^{93} + 10 q^{95} - 19 q^{97} - 37 q^{99}+O(q^{100}) $ 5 * q - q^3 - 5 * q^5 + 2 * q^7 + 6 * q^9 - 8 * q^11 - 4 * q^13 + q^15 - q^17 - 10 * q^19 + 5 * q^21 - 4 * q^23 + 5 * q^25 - q^27 + 11 * q^29 + 3 * q^31 - 3 * q^33 - 2 * q^35 - 5 * q^37 - 18 * q^39 + 16 * q^41 - 31 * q^43 - 6 * q^45 - 5 * q^47 + 7 * q^49 - 34 * q^51 + 8 * q^53 + 8 * q^55 - 8 * q^57 - 24 * q^59 - 15 * q^61 - 19 * q^63 + 4 * q^65 - 12 * q^67 + 10 * q^69 - 5 * q^71 + 5 * q^73 - q^75 - 4 * q^77 - 4 * q^79 + 17 * q^81 - 27 * q^83 + q^85 + 4 * q^87 + 16 * q^89 - 26 * q^91 + 14 * q^93 + 10 * q^95 - 19 * q^97 - 37 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−0.612608	−0.353690	−0.176845	−	0.984239i	$-0.556589\pi$
$3$	−0.612608	−0.353690	−0.176845	+	0.984239i	$0.556589\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.03374	−1.14665	−0.573323	−	0.819329i	$-0.694346\pi$
$7$	−3.03374	−1.14665	−0.573323	+	0.819329i	$0.694346\pi$
$8$	0	0
$9$	−2.62471	−0.874904
$10$	0	0
$11$	0.741603	0.223602	0.111801	−	0.993731i	$-0.464338\pi$
$11$	0.741603	0.223602	0.111801	+	0.993731i	$0.464338\pi$
$12$	0	0
$13$	4.52946	1.25625	0.628123	−	0.778114i	$-0.283824\pi$
$13$	4.52946	1.25625	0.628123	+	0.778114i	$0.283824\pi$
$14$	0	0
$15$	0.612608	0.158175
$16$	0	0
$17$	2.10832	0.511344	0.255672	−	0.966764i	$-0.417703\pi$
$17$	2.10832	0.511344	0.255672	+	0.966764i	$0.417703\pi$
$18$	0	0
$19$	3.77534	0.866123	0.433062	−	0.901364i	$-0.357433\pi$
$19$	3.77534	0.866123	0.433062	+	0.901364i	$0.357433\pi$
$20$	0	0
$21$	1.85849	0.405557
$22$	0	0
$23$	−5.48321	−1.14333	−0.571664	−	0.820488i	$-0.693702\pi$
$23$	−5.48321	−1.14333	−0.571664	+	0.820488i	$0.693702\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.44575	0.663134
$28$	0	0
$29$	9.90434	1.83919	0.919595	−	0.392869i	$-0.128517\pi$
$29$	9.90434	1.83919	0.919595	+	0.392869i	$0.128517\pi$
$30$	0	0
$31$	4.71327	0.846528	0.423264	−	0.906006i	$-0.360884\pi$
$31$	4.71327	0.846528	0.423264	+	0.906006i	$0.360884\pi$
$32$	0	0
$33$	−0.454312	−0.0790856
$34$	0	0
$35$	3.03374	0.512796
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	−2.77478	−0.444321
$40$	0	0
$41$	−2.99959	−0.468458	−0.234229	−	0.972182i	$-0.575256\pi$
$41$	−2.99959	−0.468458	−0.234229	+	0.972182i	$0.575256\pi$
$42$	0	0
$43$	−12.1873	−1.85855	−0.929277	−	0.369385i	$-0.879568\pi$
$43$	−12.1873	−1.85855	−0.929277	+	0.369385i	$0.879568\pi$
$44$	0	0
$45$	2.62471	0.391269
$46$	0	0
$47$	−5.73788	−0.836956	−0.418478	−	0.908227i	$-0.637436\pi$
$47$	−5.73788	−0.836956	−0.418478	+	0.908227i	$0.637436\pi$
$48$	0	0
$49$	2.20358	0.314797
$50$	0	0
$51$	−1.29158	−0.180857
$52$	0	0
$53$	9.09209	1.24890	0.624448	−	0.781067i	$-0.285324\pi$
$53$	9.09209	1.24890	0.624448	+	0.781067i	$0.285324\pi$
$54$	0	0
$55$	−0.741603	−0.0999977
$56$	0	0
$57$	−2.31281	−0.306339
$58$	0	0
$59$	−5.06692	−0.659657	−0.329828	−	0.944041i	$-0.606991\pi$
$59$	−5.06692	−0.659657	−0.329828	+	0.944041i	$0.606991\pi$
$60$	0	0
$61$	−15.0296	−1.92434	−0.962172	−	0.272442i	$-0.912169\pi$
$61$	−15.0296	−1.92434	−0.962172	+	0.272442i	$0.912169\pi$
$62$	0	0
$63$	7.96269	1.00320
$64$	0	0
$65$	−4.52946	−0.561810
$66$	0	0
$67$	−3.55869	−0.434764	−0.217382	−	0.976087i	$-0.569752\pi$
$67$	−3.55869	−0.434764	−0.217382	+	0.976087i	$0.569752\pi$
$68$	0	0
$69$	3.35906	0.404383
$70$	0	0
$71$	−10.1542	−1.20508	−0.602539	−	0.798089i	$-0.705844\pi$
$71$	−10.1542	−1.20508	−0.602539	+	0.798089i	$0.705844\pi$
$72$	0	0
$73$	−8.15417	−0.954373	−0.477187	−	0.878802i	$-0.658343\pi$
$73$	−8.15417	−0.954373	−0.477187	+	0.878802i	$0.658343\pi$
$74$	0	0
$75$	−0.612608	−0.0707379
$76$	0	0
$77$	−2.24983	−0.256392
$78$	0	0
$79$	2.00913	0.226044	0.113022	−	0.993592i	$-0.463947\pi$
$79$	2.00913	0.226044	0.113022	+	0.993592i	$0.463947\pi$
$80$	0	0
$81$	5.76324	0.640360
$82$	0	0
$83$	−11.3210	−1.24264	−0.621322	−	0.783555i	$-0.713404\pi$
$83$	−11.3210	−1.24264	−0.621322	+	0.783555i	$0.713404\pi$
$84$	0	0
$85$	−2.10832	−0.228680
$86$	0	0
$87$	−6.06748	−0.650502
$88$	0	0
$89$	3.26656	0.346254	0.173127	−	0.984899i	$-0.444613\pi$
$89$	3.26656	0.346254	0.173127	+	0.984899i	$0.444613\pi$
$90$	0	0
$91$	−13.7412	−1.44047
$92$	0	0
$93$	−2.88739	−0.299408
$94$	0	0
$95$	−3.77534	−0.387342
$96$	0	0
$97$	2.18735	0.222092	0.111046	−	0.993815i	$-0.464580\pi$
$97$	2.18735	0.222092	0.111046	+	0.993815i	$0.464580\pi$
$98$	0	0
$99$	−1.94649	−0.195630
$100$	0	0
$101$	−1.62471	−0.161665	−0.0808324	−	0.996728i	$-0.525758\pi$
$101$	−1.62471	−0.161665	−0.0808324	+	0.996728i	$0.525758\pi$
$102$	0	0
$103$	5.74119	0.565697	0.282848	−	0.959165i	$-0.408721\pi$
$103$	5.74119	0.565697	0.282848	+	0.959165i	$0.408721\pi$
$104$	0	0
$105$	−1.85849	−0.181371
$106$	0	0
$107$	0.699296	0.0676035	0.0338017	−	0.999429i	$-0.489239\pi$
$107$	0.699296	0.0676035	0.0338017	+	0.999429i	$0.489239\pi$
$108$	0	0
$109$	−6.87157	−0.658177	−0.329088	−	0.944299i	$-0.606741\pi$
$109$	−6.87157	−0.658177	−0.329088	+	0.944299i	$0.606741\pi$
$110$	0	0
$111$	0.612608	0.0581462
$112$	0	0
$113$	9.14110	0.859922	0.429961	−	0.902847i	$-0.358527\pi$
$113$	9.14110	0.859922	0.429961	+	0.902847i	$0.358527\pi$
$114$	0	0
$115$	5.48321	0.511312
$116$	0	0
$117$	−11.8885	−1.09909
$118$	0	0
$119$	−6.39611	−0.586330
$120$	0	0
$121$	−10.4500	−0.950002
$122$	0	0
$123$	1.83758	0.165689
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	1.62118	0.143856	0.0719280	−	0.997410i	$-0.477085\pi$
$127$	1.62118	0.143856	0.0719280	+	0.997410i	$0.477085\pi$
$128$	0	0
$129$	7.46607	0.657351
$130$	0	0
$131$	−18.0162	−1.57408	−0.787041	−	0.616900i	$-0.788388\pi$
$131$	−18.0162	−1.57408	−0.787041	+	0.616900i	$0.788388\pi$
$132$	0	0
$133$	−11.4534	−0.993137
$134$	0	0
$135$	−3.44575	−0.296563
$136$	0	0
$137$	−16.8251	−1.43747	−0.718734	−	0.695285i	$-0.755278\pi$
$137$	−16.8251	−1.43747	−0.718734	+	0.695285i	$0.755278\pi$
$138$	0	0
$139$	−6.70414	−0.568638	−0.284319	−	0.958730i	$-0.591767\pi$
$139$	−6.70414	−0.568638	−0.284319	+	0.958730i	$0.591767\pi$
$140$	0	0
$141$	3.51508	0.296023
$142$	0	0
$143$	3.35906	0.280899
$144$	0	0
$145$	−9.90434	−0.822510
$146$	0	0
$147$	−1.34993	−0.111340
$148$	0	0
$149$	19.5922	1.60506	0.802530	−	0.596612i	$-0.203487\pi$
$149$	19.5922	1.60506	0.802530	+	0.596612i	$0.203487\pi$
$150$	0	0
$151$	−7.55069	−0.614466	−0.307233	−	0.951634i	$-0.599403\pi$
$151$	−7.55069	−0.614466	−0.307233	+	0.951634i	$0.599403\pi$
$152$	0	0
$153$	−5.53374	−0.447377
$154$	0	0
$155$	−4.71327	−0.378579
$156$	0	0
$157$	−8.60776	−0.686974	−0.343487	−	0.939157i	$-0.611608\pi$
$157$	−8.60776	−0.686974	−0.343487	+	0.939157i	$0.611608\pi$
$158$	0	0
$159$	−5.56989	−0.441721
$160$	0	0
$161$	16.6346	1.31099
$162$	0	0
$163$	−4.07064	−0.318837	−0.159419	−	0.987211i	$-0.550962\pi$
$163$	−4.07064	−0.318837	−0.159419	+	0.987211i	$0.550962\pi$
$164$	0	0
$165$	0.454312	0.0353682
$166$	0	0
$167$	11.5507	0.893819	0.446909	−	0.894579i	$-0.352525\pi$
$167$	11.5507	0.893819	0.446909	+	0.894579i	$0.352525\pi$
$168$	0	0
$169$	7.51598	0.578152
$170$	0	0
$171$	−9.90918	−0.757774
$172$	0	0
$173$	−3.60889	−0.274379	−0.137189	−	0.990545i	$-0.543807\pi$
$173$	−3.60889	−0.274379	−0.137189	+	0.990545i	$0.543807\pi$
$174$	0	0
$175$	−3.03374	−0.229329
$176$	0	0
$177$	3.10404	0.233314
$178$	0	0
$179$	10.4088	0.777994	0.388997	−	0.921239i	$-0.372822\pi$
$179$	10.4088	0.777994	0.388997	+	0.921239i	$0.372822\pi$
$180$	0	0
$181$	−13.3417	−0.991681	−0.495840	−	0.868414i	$-0.665140\pi$
$181$	−13.3417	−0.991681	−0.495840	+	0.868414i	$0.665140\pi$
$182$	0	0
$183$	9.20727	0.680621
$184$	0	0
$185$	1.00000	0.0735215
$186$	0	0
$187$	1.56354	0.114337
$188$	0	0
$189$	−10.4535	−0.760380
$190$	0	0
$191$	13.4046	0.969920	0.484960	−	0.874536i	$-0.338834\pi$
$191$	13.4046	0.969920	0.484960	+	0.874536i	$0.338834\pi$
$192$	0	0
$193$	7.68453	0.553144	0.276572	−	0.960993i	$-0.410802\pi$
$193$	7.68453	0.553144	0.276572	+	0.960993i	$0.410802\pi$
$194$	0	0
$195$	2.77478	0.198706
$196$	0	0
$197$	1.14448	0.0815409	0.0407705	−	0.999169i	$-0.487019\pi$
$197$	1.14448	0.0815409	0.0407705	+	0.999169i	$0.487019\pi$
$198$	0	0
$199$	22.9017	1.62346	0.811731	−	0.584032i	$-0.198526\pi$
$199$	22.9017	1.62346	0.811731	+	0.584032i	$0.198526\pi$
$200$	0	0
$201$	2.18009	0.153771
$202$	0	0
$203$	−30.0472	−2.10890
$204$	0	0
$205$	2.99959	0.209501
$206$	0	0
$207$	14.3918	1.00030
$208$	0	0
$209$	2.79980	0.193667
$210$	0	0
$211$	19.1174	1.31610	0.658049	−	0.752975i	$-0.271382\pi$
$211$	19.1174	1.31610	0.658049	+	0.752975i	$0.271382\pi$
$212$	0	0
$213$	6.22053	0.426224
$214$	0	0
$215$	12.1873	0.831170
$216$	0	0
$217$	−14.2988	−0.970668
$218$	0	0
$219$	4.99531	0.337552
$220$	0	0
$221$	9.54957	0.642373
$222$	0	0
$223$	−1.22618	−0.0821114	−0.0410557	−	0.999157i	$-0.513072\pi$
$223$	−1.22618	−0.0821114	−0.0410557	+	0.999157i	$0.513072\pi$
$224$	0	0
$225$	−2.62471	−0.174981
$226$	0	0
$227$	14.0151	0.930214	0.465107	−	0.885254i	$-0.346016\pi$
$227$	14.0151	0.930214	0.465107	+	0.885254i	$0.346016\pi$
$228$	0	0
$229$	14.1049	0.932081	0.466040	−	0.884763i	$-0.345680\pi$
$229$	14.1049	0.932081	0.466040	+	0.884763i	$0.345680\pi$
$230$	0	0
$231$	1.37826	0.0906832
$232$	0	0
$233$	−28.9601	−1.89724	−0.948619	−	0.316420i	$-0.897519\pi$
$233$	−28.9601	−1.89724	−0.948619	+	0.316420i	$0.897519\pi$
$234$	0	0
$235$	5.73788	0.374298
$236$	0	0
$237$	−1.23081	−0.0799496
$238$	0	0
$239$	4.64691	0.300584	0.150292	−	0.988642i	$-0.451979\pi$
$239$	4.64691	0.300584	0.150292	+	0.988642i	$0.451979\pi$
$240$	0	0
$241$	20.7751	1.33824	0.669120	−	0.743154i	$-0.266671\pi$
$241$	20.7751	1.33824	0.669120	+	0.743154i	$0.266671\pi$
$242$	0	0
$243$	−13.8678	−0.889623
$244$	0	0
$245$	−2.20358	−0.140782
$246$	0	0
$247$	17.1003	1.08806
$248$	0	0
$249$	6.93536	0.439510
$250$	0	0
$251$	13.1999	0.833173	0.416587	−	0.909096i	$-0.363226\pi$
$251$	13.1999	0.833173	0.416587	+	0.909096i	$0.363226\pi$
$252$	0	0
$253$	−4.06636	−0.255650
$254$	0	0
$255$	1.29158	0.0808817
$256$	0	0
$257$	−22.5286	−1.40530	−0.702649	−	0.711537i	$-0.748000\pi$
$257$	−22.5286	−1.40530	−0.702649	+	0.711537i	$0.748000\pi$
$258$	0	0
$259$	3.03374	0.188507
$260$	0	0
$261$	−25.9960	−1.60911
$262$	0	0
$263$	−32.0110	−1.97388	−0.986940	−	0.161086i	$-0.948500\pi$
$263$	−32.0110	−1.97388	−0.986940	+	0.161086i	$0.948500\pi$
$264$	0	0
$265$	−9.09209	−0.558523
$266$	0	0
$267$	−2.00112	−0.122467
$268$	0	0
$269$	−14.4306	−0.879851	−0.439926	−	0.898034i	$-0.644995\pi$
$269$	−14.4306	−0.879851	−0.439926	+	0.898034i	$0.644995\pi$
$270$	0	0
$271$	−9.63819	−0.585478	−0.292739	−	0.956192i	$-0.594567\pi$
$271$	−9.63819	−0.585478	−0.292739	+	0.956192i	$0.594567\pi$
$272$	0	0
$273$	8.41797	0.509479
$274$	0	0
$275$	0.741603	0.0447203
$276$	0	0
$277$	0.404492	0.0243036	0.0121518	−	0.999926i	$-0.496132\pi$
$277$	0.404492	0.0243036	0.0121518	+	0.999926i	$0.496132\pi$
$278$	0	0
$279$	−12.3710	−0.740631
$280$	0	0
$281$	4.65175	0.277500	0.138750	−	0.990327i	$-0.455691\pi$
$281$	4.65175	0.277500	0.138750	+	0.990327i	$0.455691\pi$
$282$	0	0
$283$	−27.5991	−1.64060	−0.820298	−	0.571936i	$-0.806192\pi$
$283$	−27.5991	−1.64060	−0.820298	+	0.571936i	$0.806192\pi$
$284$	0	0
$285$	2.31281	0.136999
$286$	0	0
$287$	9.09998	0.537155
$288$	0	0
$289$	−12.5550	−0.738527
$290$	0	0
$291$	−1.33999	−0.0785515
$292$	0	0
$293$	−29.7884	−1.74026	−0.870128	−	0.492826i	$-0.835964\pi$
$293$	−29.7884	−1.74026	−0.870128	+	0.492826i	$0.835964\pi$
$294$	0	0
$295$	5.06692	0.295008
$296$	0	0
$297$	2.55537	0.148278
$298$	0	0
$299$	−24.8359	−1.43630
$300$	0	0
$301$	36.9732	2.13110
$302$	0	0
$303$	0.995312	0.0571792
$304$	0	0
$305$	15.0296	0.860593
$306$	0	0
$307$	−30.5372	−1.74285	−0.871424	−	0.490530i	$-0.836803\pi$
$307$	−30.5372	−1.74285	−0.871424	+	0.490530i	$0.836803\pi$
$308$	0	0
$309$	−3.51710	−0.200081
$310$	0	0
$311$	33.3363	1.89033	0.945163	−	0.326599i	$-0.105903\pi$
$311$	33.3363	1.89033	0.945163	+	0.326599i	$0.105903\pi$
$312$	0	0
$313$	−14.9835	−0.846920	−0.423460	−	0.905915i	$-0.639185\pi$
$313$	−14.9835	−0.846920	−0.423460	+	0.905915i	$0.639185\pi$
$314$	0	0
$315$	−7.96269	−0.448647
$316$	0	0
$317$	−18.9633	−1.06508	−0.532541	−	0.846404i	$-0.678763\pi$
$317$	−18.9633	−1.06508	−0.532541	+	0.846404i	$0.678763\pi$
$318$	0	0
$319$	7.34508	0.411246
$320$	0	0
$321$	−0.428394	−0.0239106
$322$	0	0
$323$	7.95965	0.442887
$324$	0	0
$325$	4.52946	0.251249
$326$	0	0
$327$	4.20958	0.232790
$328$	0	0
$329$	17.4072	0.959692
$330$	0	0
$331$	−26.1008	−1.43463	−0.717315	−	0.696749i	$-0.754629\pi$
$331$	−26.1008	−1.43463	−0.717315	+	0.696749i	$0.754629\pi$
$332$	0	0
$333$	2.62471	0.143833
$334$	0	0
$335$	3.55869	0.194432
$336$	0	0
$337$	−6.26976	−0.341535	−0.170768	−	0.985311i	$-0.554625\pi$
$337$	−6.26976	−0.341535	−0.170768	+	0.985311i	$0.554625\pi$
$338$	0	0
$339$	−5.59991	−0.304146
$340$	0	0
$341$	3.49537	0.189285
$342$	0	0
$343$	14.5511	0.785685
$344$	0	0
$345$	−3.35906	−0.180846
$346$	0	0
$347$	20.3083	1.09021	0.545104	−	0.838368i	$-0.316490\pi$
$347$	20.3083	1.09021	0.545104	+	0.838368i	$0.316490\pi$
$348$	0	0
$349$	24.3381	1.30279	0.651395	−	0.758739i	$-0.274184\pi$
$349$	24.3381	1.30279	0.651395	+	0.758739i	$0.274184\pi$
$350$	0	0
$351$	15.6074	0.833059
$352$	0	0
$353$	−11.6907	−0.622235	−0.311118	−	0.950371i	$-0.600703\pi$
$353$	−11.6907	−0.622235	−0.311118	+	0.950371i	$0.600703\pi$
$354$	0	0
$355$	10.1542	0.538927
$356$	0	0
$357$	3.91831	0.207379
$358$	0	0
$359$	−0.0117664	−0.000621007 0	−0.000310504	−	1.00000i	$-0.500099\pi$
$359$	−0.0117664	−0.000621007 0	−0.000310504		1.00000i	$0.500099\pi$
$360$	0	0
$361$	−4.74679	−0.249831
$362$	0	0
$363$	6.40177	0.336006
$364$	0	0
$365$	8.15417	0.426809
$366$	0	0
$367$	−18.1651	−0.948212	−0.474106	−	0.880468i	$-0.657229\pi$
$367$	−18.1651	−0.948212	−0.474106	+	0.880468i	$0.657229\pi$
$368$	0	0
$369$	7.87306	0.409855
$370$	0	0
$371$	−27.5830	−1.43204
$372$	0	0
$373$	0.996431	0.0515932	0.0257966	−	0.999667i	$-0.491788\pi$
$373$	0.996431	0.0515932	0.0257966	+	0.999667i	$0.491788\pi$
$374$	0	0
$375$	0.612608	0.0316350
$376$	0	0
$377$	44.8613	2.31047
$378$	0	0
$379$	10.6798	0.548586	0.274293	−	0.961646i	$-0.411556\pi$
$379$	10.6798	0.548586	0.274293	+	0.961646i	$0.411556\pi$
$380$	0	0
$381$	−0.993146	−0.0508804
$382$	0	0
$383$	6.25055	0.319388	0.159694	−	0.987167i	$-0.448949\pi$
$383$	6.25055	0.319388	0.159694	+	0.987167i	$0.448949\pi$
$384$	0	0
$385$	2.24983	0.114662
$386$	0	0
$387$	31.9883	1.62605
$388$	0	0
$389$	−34.6153	−1.75507	−0.877533	−	0.479516i	$-0.840812\pi$
$389$	−34.6153	−1.75507	−0.877533	+	0.479516i	$0.840812\pi$
$390$	0	0
$391$	−11.5604	−0.584634
$392$	0	0
$393$	11.0369	0.556737
$394$	0	0
$395$	−2.00913	−0.101090
$396$	0	0
$397$	−37.5377	−1.88396	−0.941982	−	0.335662i	$-0.891040\pi$
$397$	−37.5377	−1.88396	−0.941982	+	0.335662i	$0.891040\pi$
$398$	0	0
$399$	7.01645	0.351262
$400$	0	0
$401$	−16.6338	−0.830653	−0.415326	−	0.909673i	$-0.636333\pi$
$401$	−16.6338	−0.830653	−0.415326	+	0.909673i	$0.636333\pi$
$402$	0	0
$403$	21.3485	1.06345
$404$	0	0
$405$	−5.76324	−0.286378
$406$	0	0
$407$	−0.741603	−0.0367599
$408$	0	0
$409$	−7.72780	−0.382115	−0.191058	−	0.981579i	$-0.561192\pi$
$409$	−7.72780	−0.382115	−0.191058	+	0.981579i	$0.561192\pi$
$410$	0	0
$411$	10.3072	0.508417
$412$	0	0
$413$	15.3717	0.756393
$414$	0	0
$415$	11.3210	0.555728
$416$	0	0
$417$	4.10701	0.201121
$418$	0	0
$419$	−28.2216	−1.37872	−0.689359	−	0.724420i	$-0.742108\pi$
$419$	−28.2216	−1.37872	−0.689359	+	0.724420i	$0.742108\pi$
$420$	0	0
$421$	−25.9504	−1.26475	−0.632373	−	0.774664i	$-0.717919\pi$
$421$	−25.9504	−1.26475	−0.632373	+	0.774664i	$0.717919\pi$
$422$	0	0
$423$	15.0603	0.732256
$424$	0	0
$425$	2.10832	0.102269
$426$	0	0
$427$	45.5959	2.20654
$428$	0	0
$429$	−2.05779	−0.0993509
$430$	0	0
$431$	28.7819	1.38637	0.693187	−	0.720758i	$-0.256206\pi$
$431$	28.7819	1.38637	0.693187	+	0.720758i	$0.256206\pi$
$432$	0	0
$433$	−28.3324	−1.36157	−0.680784	−	0.732484i	$-0.738361\pi$
$433$	−28.3324	−1.36157	−0.680784	+	0.732484i	$0.738361\pi$
$434$	0	0
$435$	6.06748	0.290913
$436$	0	0
$437$	−20.7010	−0.990262
$438$	0	0
$439$	9.30386	0.444049	0.222025	−	0.975041i	$-0.428733\pi$
$439$	9.30386	0.444049	0.222025	+	0.975041i	$0.428733\pi$
$440$	0	0
$441$	−5.78376	−0.275417
$442$	0	0
$443$	−21.6446	−1.02836	−0.514182	−	0.857681i	$-0.671904\pi$
$443$	−21.6446	−1.02836	−0.514182	+	0.857681i	$0.671904\pi$
$444$	0	0
$445$	−3.26656	−0.154850
$446$	0	0
$447$	−12.0024	−0.567693
$448$	0	0
$449$	20.8684	0.984841	0.492420	−	0.870357i	$-0.336112\pi$
$449$	20.8684	0.984841	0.492420	+	0.870357i	$0.336112\pi$
$450$	0	0
$451$	−2.22451	−0.104748
$452$	0	0
$453$	4.62561	0.217330
$454$	0	0
$455$	13.7412	0.644197
$456$	0	0
$457$	−6.63473	−0.310359	−0.155180	−	0.987886i	$-0.549596\pi$
$457$	−6.63473	−0.310359	−0.155180	+	0.987886i	$0.549596\pi$
$458$	0	0
$459$	7.26475	0.339090
$460$	0	0
$461$	12.6176	0.587660	0.293830	−	0.955858i	$-0.405070\pi$
$461$	12.6176	0.587660	0.293830	+	0.955858i	$0.405070\pi$
$462$	0	0
$463$	−25.7594	−1.19714	−0.598572	−	0.801069i	$-0.704265\pi$
$463$	−25.7594	−1.19714	−0.598572	+	0.801069i	$0.704265\pi$
$464$	0	0
$465$	2.88739	0.133899
$466$	0	0
$467$	8.83023	0.408614	0.204307	−	0.978907i	$-0.434506\pi$
$467$	8.83023	0.408614	0.204307	+	0.978907i	$0.434506\pi$
$468$	0	0
$469$	10.7962	0.498520
$470$	0	0
$471$	5.27319	0.242976
$472$	0	0
$473$	−9.03817	−0.415576
$474$	0	0
$475$	3.77534	0.173225
$476$	0	0
$477$	−23.8641	−1.09266
$478$	0	0
$479$	20.8141	0.951019	0.475509	−	0.879711i	$-0.342264\pi$
$479$	20.8141	0.951019	0.475509	+	0.879711i	$0.342264\pi$
$480$	0	0
$481$	−4.52946	−0.206525
$482$	0	0
$483$	−10.1905	−0.463684
$484$	0	0
$485$	−2.18735	−0.0993224
$486$	0	0
$487$	16.3103	0.739089	0.369544	−	0.929213i	$-0.379514\pi$
$487$	16.3103	0.739089	0.369544	+	0.929213i	$0.379514\pi$
$488$	0	0
$489$	2.49371	0.112769
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	20.8816	0.940458
$494$	0	0
$495$	1.94649	0.0874884
$496$	0	0
$497$	30.8051	1.38180
$498$	0	0
$499$	16.4581	0.736764	0.368382	−	0.929674i	$-0.379912\pi$
$499$	16.4581	0.736764	0.368382	+	0.929674i	$0.379912\pi$
$500$	0	0
$501$	−7.07605	−0.316134
$502$	0	0
$503$	−26.9015	−1.19948	−0.599739	−	0.800196i	$-0.704729\pi$
$503$	−26.9015	−1.19948	−0.599739	+	0.800196i	$0.704729\pi$
$504$	0	0
$505$	1.62471	0.0722987
$506$	0	0
$507$	−4.60435	−0.204486
$508$	0	0
$509$	32.1542	1.42521	0.712604	−	0.701566i	$-0.247516\pi$
$509$	32.1542	1.42521	0.712604	+	0.701566i	$0.247516\pi$
$510$	0	0
$511$	24.7376	1.09433
$512$	0	0
$513$	13.0089	0.574356
$514$	0	0
$515$	−5.74119	−0.252987
$516$	0	0
$517$	−4.25523	−0.187145
$518$	0	0
$519$	2.21084	0.0970449
$520$	0	0
$521$	−10.0720	−0.441262	−0.220631	−	0.975357i	$-0.570812\pi$
$521$	−10.0720	−0.441262	−0.220631	+	0.975357i	$0.570812\pi$
$522$	0	0
$523$	−9.62449	−0.420849	−0.210425	−	0.977610i	$-0.567485\pi$
$523$	−9.62449	−0.420849	−0.210425	+	0.977610i	$0.567485\pi$
$524$	0	0
$525$	1.85849	0.0811114
$526$	0	0
$527$	9.93710	0.432867
$528$	0	0
$529$	7.06554	0.307198
$530$	0	0
$531$	13.2992	0.577136
$532$	0	0
$533$	−13.5865	−0.588498
$534$	0	0
$535$	−0.699296	−0.0302332
$536$	0	0
$537$	−6.37655	−0.275168
$538$	0	0
$539$	1.63418	0.0703891
$540$	0	0
$541$	−33.1845	−1.42671	−0.713356	−	0.700801i	$-0.752826\pi$
$541$	−33.1845	−1.42671	−0.713356	+	0.700801i	$0.752826\pi$
$542$	0	0
$543$	8.17324	0.350747
$544$	0	0
$545$	6.87157	0.294346
$546$	0	0
$547$	21.6205	0.924426	0.462213	−	0.886769i	$-0.347056\pi$
$547$	21.6205	0.924426	0.462213	+	0.886769i	$0.347056\pi$
$548$	0	0
$549$	39.4484	1.68362
$550$	0	0
$551$	37.3923	1.59296
$552$	0	0
$553$	−6.09517	−0.259193
$554$	0	0
$555$	−0.612608	−0.0260038
$556$	0	0
$557$	41.0477	1.73925	0.869623	−	0.493717i	$-0.164362\pi$
$557$	41.0477	1.73925	0.869623	+	0.493717i	$0.164362\pi$
$558$	0	0
$559$	−55.2021	−2.33480
$560$	0	0
$561$	−0.957838	−0.0404399
$562$	0	0
$563$	17.6042	0.741928	0.370964	−	0.928647i	$-0.379027\pi$
$563$	17.6042	0.741928	0.370964	+	0.928647i	$0.379027\pi$
$564$	0	0
$565$	−9.14110	−0.384569
$566$	0	0
$567$	−17.4842	−0.734266
$568$	0	0
$569$	25.1614	1.05482	0.527411	−	0.849611i	$-0.323163\pi$
$569$	25.1614	1.05482	0.527411	+	0.849611i	$0.323163\pi$
$570$	0	0
$571$	3.96600	0.165972	0.0829860	−	0.996551i	$-0.473554\pi$
$571$	3.96600	0.165972	0.0829860	+	0.996551i	$0.473554\pi$
$572$	0	0
$573$	−8.21174	−0.343051
$574$	0	0
$575$	−5.48321	−0.228665
$576$	0	0
$577$	8.56886	0.356726	0.178363	−	0.983965i	$-0.442920\pi$
$577$	8.56886	0.356726	0.178363	+	0.983965i	$0.442920\pi$
$578$	0	0
$579$	−4.70761	−0.195641
$580$	0	0
$581$	34.3451	1.42487
$582$	0	0
$583$	6.74272	0.279255
$584$	0	0
$585$	11.8885	0.491530
$586$	0	0
$587$	6.00316	0.247777	0.123888	−	0.992296i	$-0.460463\pi$
$587$	6.00316	0.247777	0.123888	+	0.992296i	$0.460463\pi$
$588$	0	0
$589$	17.7942	0.733198
$590$	0	0
$591$	−0.701119	−0.0288402
$592$	0	0
$593$	23.8712	0.980271	0.490135	−	0.871646i	$-0.336947\pi$
$593$	23.8712	0.980271	0.490135	+	0.871646i	$0.336947\pi$
$594$	0	0
$595$	6.39611	0.262215
$596$	0	0
$597$	−14.0298	−0.574201
$598$	0	0
$599$	−9.38020	−0.383265	−0.191632	−	0.981467i	$-0.561378\pi$
$599$	−9.38020	−0.383265	−0.191632	+	0.981467i	$0.561378\pi$
$600$	0	0
$601$	−1.54907	−0.0631880	−0.0315940	−	0.999501i	$-0.510058\pi$
$601$	−1.54907	−0.0631880	−0.0315940	+	0.999501i	$0.510058\pi$
$602$	0	0
$603$	9.34054	0.380376
$604$	0	0
$605$	10.4500	0.424854
$606$	0	0
$607$	−15.9937	−0.649163	−0.324582	−	0.945858i	$-0.605224\pi$
$607$	−15.9937	−0.649163	−0.324582	+	0.945858i	$0.605224\pi$
$608$	0	0
$609$	18.4072	0.745896
$610$	0	0
$611$	−25.9895	−1.05142
$612$	0	0
$613$	34.6507	1.39953	0.699764	−	0.714374i	$-0.253289\pi$
$613$	34.6507	1.39953	0.699764	+	0.714374i	$0.253289\pi$
$614$	0	0
$615$	−1.83758	−0.0740982
$616$	0	0
$617$	23.0379	0.927471	0.463735	−	0.885974i	$-0.346509\pi$
$617$	23.0379	0.927471	0.463735	+	0.885974i	$0.346509\pi$
$618$	0	0
$619$	17.3743	0.698332	0.349166	−	0.937061i	$-0.386465\pi$
$619$	17.3743	0.698332	0.349166	+	0.937061i	$0.386465\pi$
$620$	0	0
$621$	−18.8937	−0.758179
$622$	0	0
$623$	−9.90988	−0.397031
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−1.71518	−0.0684979
$628$	0	0
$629$	−2.10832	−0.0840644
$630$	0	0
$631$	−34.1204	−1.35831	−0.679156	−	0.733994i	$-0.737654\pi$
$631$	−34.1204	−1.35831	−0.679156	+	0.733994i	$0.737654\pi$
$632$	0	0
$633$	−11.7115	−0.465490
$634$	0	0
$635$	−1.62118	−0.0643344
$636$	0	0
$637$	9.98102	0.395462
$638$	0	0
$639$	26.6518	1.05433
$640$	0	0
$641$	−8.35418	−0.329970	−0.164985	−	0.986296i	$-0.552758\pi$
$641$	−8.35418	−0.329970	−0.164985	+	0.986296i	$0.552758\pi$
$642$	0	0
$643$	3.84668	0.151698	0.0758491	−	0.997119i	$-0.475833\pi$
$643$	3.84668	0.151698	0.0758491	+	0.997119i	$0.475833\pi$
$644$	0	0
$645$	−7.46607	−0.293976
$646$	0	0
$647$	28.2699	1.11141	0.555703	−	0.831381i	$-0.312449\pi$
$647$	28.2699	1.11141	0.555703	+	0.831381i	$0.312449\pi$
$648$	0	0
$649$	−3.75764	−0.147500
$650$	0	0
$651$	8.75959	0.343315
$652$	0	0
$653$	18.2099	0.712608	0.356304	−	0.934370i	$-0.384037\pi$
$653$	18.2099	0.712608	0.356304	+	0.934370i	$0.384037\pi$
$654$	0	0
$655$	18.0162	0.703951
$656$	0	0
$657$	21.4023	0.834985
$658$	0	0
$659$	−21.8802	−0.852330	−0.426165	−	0.904645i	$-0.640136\pi$
$659$	−21.8802	−0.852330	−0.426165	+	0.904645i	$0.640136\pi$
$660$	0	0
$661$	−2.90321	−0.112922	−0.0564609	−	0.998405i	$-0.517982\pi$
$661$	−2.90321	−0.112922	−0.0564609	+	0.998405i	$0.517982\pi$
$662$	0	0
$663$	−5.85014	−0.227201
$664$	0	0
$665$	11.4534	0.444144
$666$	0	0
$667$	−54.3075	−2.10280
$668$	0	0
$669$	0.751171	0.0290420
$670$	0	0
$671$	−11.1460	−0.430287
$672$	0	0
$673$	17.1698	0.661847	0.330924	−	0.943658i	$-0.392640\pi$
$673$	17.1698	0.661847	0.330924	+	0.943658i	$0.392640\pi$
$674$	0	0
$675$	3.44575	0.132627
$676$	0	0
$677$	21.6974	0.833899	0.416950	−	0.908930i	$-0.363099\pi$
$677$	21.6974	0.833899	0.416950	+	0.908930i	$0.363099\pi$
$678$	0	0
$679$	−6.63585	−0.254660
$680$	0	0
$681$	−8.58576	−0.329007
$682$	0	0
$683$	31.2545	1.19592	0.597959	−	0.801527i	$-0.295978\pi$
$683$	31.2545	1.19592	0.597959	+	0.801527i	$0.295978\pi$
$684$	0	0
$685$	16.8251	0.642855
$686$	0	0
$687$	−8.64081	−0.329667
$688$	0	0
$689$	41.1822	1.56892
$690$	0	0
$691$	−47.6654	−1.81328	−0.906638	−	0.421910i	$-0.861360\pi$
$691$	−47.6654	−1.81328	−0.906638	+	0.421910i	$0.861360\pi$
$692$	0	0
$693$	5.90515	0.224318
$694$	0	0
$695$	6.70414	0.254303
$696$	0	0
$697$	−6.32411	−0.239543
$698$	0	0
$699$	17.7412	0.671034
$700$	0	0
$701$	20.8184	0.786299	0.393149	−	0.919475i	$-0.371386\pi$
$701$	20.8184	0.786299	0.393149	+	0.919475i	$0.371386\pi$
$702$	0	0
$703$	−3.77534	−0.142390
$704$	0	0
$705$	−3.51508	−0.132385
$706$	0	0
$707$	4.92895	0.185372
$708$	0	0
$709$	45.8047	1.72023	0.860116	−	0.510099i	$-0.170391\pi$
$709$	45.8047	1.72023	0.860116	+	0.510099i	$0.170391\pi$
$710$	0	0
$711$	−5.27338	−0.197767
$712$	0	0
$713$	−25.8438	−0.967859
$714$	0	0
$715$	−3.35906	−0.125622
$716$	0	0
$717$	−2.84674	−0.106313
$718$	0	0
$719$	−1.64481	−0.0613412	−0.0306706	−	0.999530i	$-0.509764\pi$
$719$	−1.64481	−0.0613412	−0.0306706	+	0.999530i	$0.509764\pi$
$720$	0	0
$721$	−17.4173	−0.648654
$722$	0	0
$723$	−12.7270	−0.473322
$724$	0	0
$725$	9.90434	0.367838
$726$	0	0
$727$	−0.550245	−0.0204075	−0.0102037	−	0.999948i	$-0.503248\pi$
$727$	−0.550245	−0.0204075	−0.0102037	+	0.999948i	$0.503248\pi$
$728$	0	0
$729$	−8.79416	−0.325710
$730$	0	0
$731$	−25.6949	−0.950360
$732$	0	0
$733$	3.06982	0.113387	0.0566933	−	0.998392i	$-0.481944\pi$
$733$	3.06982	0.113387	0.0566933	+	0.998392i	$0.481944\pi$
$734$	0	0
$735$	1.34993	0.0497930
$736$	0	0
$737$	−2.63914	−0.0972139
$738$	0	0
$739$	−11.9253	−0.438681	−0.219341	−	0.975648i	$-0.570391\pi$
$739$	−11.9253	−0.438681	−0.219341	+	0.975648i	$0.570391\pi$
$740$	0	0
$741$	−10.4758	−0.384837
$742$	0	0
$743$	−2.06651	−0.0758130	−0.0379065	−	0.999281i	$-0.512069\pi$
$743$	−2.06651	−0.0758130	−0.0379065	+	0.999281i	$0.512069\pi$
$744$	0	0
$745$	−19.5922	−0.717804
$746$	0	0
$747$	29.7144	1.08719
$748$	0	0
$749$	−2.12148	−0.0775172
$750$	0	0
$751$	2.34356	0.0855176	0.0427588	−	0.999085i	$-0.486385\pi$
$751$	2.34356	0.0855176	0.0427588	+	0.999085i	$0.486385\pi$
$752$	0	0
$753$	−8.08640	−0.294685
$754$	0	0
$755$	7.55069	0.274798
$756$	0	0
$757$	1.00068	0.0363703	0.0181851	−	0.999835i	$-0.494211\pi$
$757$	1.00068	0.0363703	0.0181851	+	0.999835i	$0.494211\pi$
$758$	0	0
$759$	2.49109	0.0904207
$760$	0	0
$761$	12.6078	0.457031	0.228516	−	0.973540i	$-0.426613\pi$
$761$	12.6078	0.457031	0.228516	+	0.973540i	$0.426613\pi$
$762$	0	0
$763$	20.8465	0.754696
$764$	0	0
$765$	5.53374	0.200073
$766$	0	0
$767$	−22.9504	−0.828691
$768$	0	0
$769$	−44.1942	−1.59368	−0.796842	−	0.604188i	$-0.793497\pi$
$769$	−44.1942	−1.59368	−0.796842	+	0.604188i	$0.793497\pi$
$770$	0	0
$771$	13.8012	0.497039
$772$	0	0
$773$	23.7096	0.852774	0.426387	−	0.904541i	$-0.359786\pi$
$773$	23.7096	0.852774	0.426387	+	0.904541i	$0.359786\pi$
$774$	0	0
$775$	4.71327	0.169306
$776$	0	0
$777$	−1.85849	−0.0666731
$778$	0	0
$779$	−11.3245	−0.405742
$780$	0	0
$781$	−7.53036	−0.269457
$782$	0	0
$783$	34.1278	1.21963
$784$	0	0
$785$	8.60776	0.307224
$786$	0	0
$787$	−35.1214	−1.25194	−0.625972	−	0.779845i	$-0.715298\pi$
$787$	−35.1214	−1.25194	−0.625972	+	0.779845i	$0.715298\pi$
$788$	0	0
$789$	19.6102	0.698141
$790$	0	0
$791$	−27.7317	−0.986026
$792$	0	0
$793$	−68.0760	−2.41745
$794$	0	0
$795$	5.56989	0.197544
$796$	0	0
$797$	−32.4235	−1.14850	−0.574250	−	0.818680i	$-0.694706\pi$
$797$	−32.4235	−1.14850	−0.574250	+	0.818680i	$0.694706\pi$
$798$	0	0
$799$	−12.0973	−0.427972
$800$	0	0
$801$	−8.57376	−0.302939
$802$	0	0
$803$	−6.04715	−0.213399
$804$	0	0
$805$	−16.6346	−0.586293
$806$	0	0
$807$	8.84033	0.311194
$808$	0	0
$809$	−36.9321	−1.29847	−0.649233	−	0.760590i	$-0.724910\pi$
$809$	−36.9321	−1.29847	−0.649233	+	0.760590i	$0.724910\pi$
$810$	0	0
$811$	−35.6897	−1.25323	−0.626617	−	0.779328i	$-0.715561\pi$
$811$	−35.6897	−1.25323	−0.626617	+	0.779328i	$0.715561\pi$
$812$	0	0
$813$	5.90444	0.207078
$814$	0	0
$815$	4.07064	0.142588
$816$	0	0
$817$	−46.0114	−1.60974
$818$	0	0
$819$	36.0667	1.26027
$820$	0	0
$821$	−6.74732	−0.235483	−0.117742	−	0.993044i	$-0.537565\pi$
$821$	−6.74732	−0.235483	−0.117742	+	0.993044i	$0.537565\pi$
$822$	0	0
$823$	8.76858	0.305654	0.152827	−	0.988253i	$-0.451162\pi$
$823$	8.76858	0.305654	0.152827	+	0.988253i	$0.451162\pi$
$824$	0	0
$825$	−0.454312	−0.0158171
$826$	0	0
$827$	−52.1993	−1.81515	−0.907573	−	0.419893i	$-0.862067\pi$
$827$	−52.1993	−1.81515	−0.907573	+	0.419893i	$0.862067\pi$
$828$	0	0
$829$	23.3559	0.811184	0.405592	−	0.914054i	$-0.367065\pi$
$829$	23.3559	0.811184	0.405592	+	0.914054i	$0.367065\pi$
$830$	0	0
$831$	−0.247795	−0.00859592
$832$	0	0
$833$	4.64586	0.160970
$834$	0	0
$835$	−11.5507	−0.399728
$836$	0	0
$837$	16.2407	0.561362
$838$	0	0
$839$	31.6674	1.09328	0.546640	−	0.837368i	$-0.315907\pi$
$839$	31.6674	1.09328	0.546640	+	0.837368i	$0.315907\pi$
$840$	0	0
$841$	69.0959	2.38262
$842$	0	0
$843$	−2.84970	−0.0981490
$844$	0	0
$845$	−7.51598	−0.258557
$846$	0	0
$847$	31.7027	1.08932
$848$	0	0
$849$	16.9074	0.580262
$850$	0	0
$851$	5.48321	0.187962
$852$	0	0
$853$	20.7204	0.709451	0.354726	−	0.934970i	$-0.384574\pi$
$853$	20.7204	0.709451	0.354726	+	0.934970i	$0.384574\pi$
$854$	0	0
$855$	9.90918	0.338887
$856$	0	0
$857$	27.2193	0.929794	0.464897	−	0.885365i	$-0.346091\pi$
$857$	27.2193	0.929794	0.464897	+	0.885365i	$0.346091\pi$
$858$	0	0
$859$	19.8824	0.678377	0.339189	−	0.940718i	$-0.389848\pi$
$859$	19.8824	0.678377	0.339189	+	0.940718i	$0.389848\pi$
$860$	0	0
$861$	−5.57473	−0.189986
$862$	0	0
$863$	−23.2454	−0.791282	−0.395641	−	0.918405i	$-0.629478\pi$
$863$	−23.2454	−0.791282	−0.395641	+	0.918405i	$0.629478\pi$
$864$	0	0
$865$	3.60889	0.122706
$866$	0	0
$867$	7.69128	0.261209
$868$	0	0
$869$	1.48997	0.0505439
$870$	0	0
$871$	−16.1189	−0.546170
$872$	0	0
$873$	−5.74116	−0.194309
$874$	0	0
$875$	3.03374	0.102559
$876$	0	0
$877$	28.7548	0.970980	0.485490	−	0.874242i	$-0.338641\pi$
$877$	28.7548	0.970980	0.485490	+	0.874242i	$0.338641\pi$
$878$	0	0
$879$	18.2486	0.615510
$880$	0	0
$881$	12.7331	0.428990	0.214495	−	0.976725i	$-0.431189\pi$
$881$	12.7331	0.428990	0.214495	+	0.976725i	$0.431189\pi$
$882$	0	0
$883$	−16.4376	−0.553169	−0.276584	−	0.960990i	$-0.589203\pi$
$883$	−16.4376	−0.553169	−0.276584	+	0.960990i	$0.589203\pi$
$884$	0	0
$885$	−3.10404	−0.104341
$886$	0	0
$887$	22.3201	0.749436	0.374718	−	0.927139i	$-0.377740\pi$
$887$	22.3201	0.749436	0.374718	+	0.927139i	$0.377740\pi$
$888$	0	0
$889$	−4.91822	−0.164952
$890$	0	0
$891$	4.27404	0.143186
$892$	0	0
$893$	−21.6625	−0.724907
$894$	0	0
$895$	−10.4088	−0.347929
$896$	0	0
$897$	15.2147	0.508004
$898$	0	0
$899$	46.6818	1.55693
$900$	0	0
$901$	19.1691	0.638615
$902$	0	0
$903$	−22.6501	−0.753749
$904$	0	0
$905$	13.3417	0.443493
$906$	0	0
$907$	−51.4031	−1.70681	−0.853406	−	0.521247i	$-0.825467\pi$
$907$	−51.4031	−1.70681	−0.853406	+	0.521247i	$0.825467\pi$
$908$	0	0
$909$	4.26440	0.141441
$910$	0	0
$911$	4.09707	0.135742	0.0678710	−	0.997694i	$-0.478379\pi$
$911$	4.09707	0.135742	0.0678710	+	0.997694i	$0.478379\pi$
$912$	0	0
$913$	−8.39571	−0.277857
$914$	0	0
$915$	−9.20727	−0.304383
$916$	0	0
$917$	54.6565	1.80492
$918$	0	0
$919$	33.6466	1.10990	0.554950	−	0.831884i	$-0.312737\pi$
$919$	33.6466	1.10990	0.554950	+	0.831884i	$0.312737\pi$
$920$	0	0
$921$	18.7073	0.616427
$922$	0	0
$923$	−45.9929	−1.51387
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	0	0
$927$	−15.0690	−0.494930
$928$	0	0
$929$	8.27308	0.271431	0.135716	−	0.990748i	$-0.456667\pi$
$929$	8.27308	0.271431	0.135716	+	0.990748i	$0.456667\pi$
$930$	0	0
$931$	8.31927	0.272653
$932$	0	0
$933$	−20.4221	−0.668589
$934$	0	0
$935$	−1.56354	−0.0511332
$936$	0	0
$937$	−23.5181	−0.768302	−0.384151	−	0.923270i	$-0.625506\pi$
$937$	−23.5181	−0.768302	−0.384151	+	0.923270i	$0.625506\pi$
$938$	0	0
$939$	9.17905	0.299547
$940$	0	0
$941$	55.7341	1.81688	0.908441	−	0.418014i	$-0.137274\pi$
$941$	55.7341	1.81688	0.908441	+	0.418014i	$0.137274\pi$
$942$	0	0
$943$	16.4474	0.535600
$944$	0	0
$945$	10.4535	0.340052
$946$	0	0
$947$	36.4274	1.18373	0.591866	−	0.806037i	$-0.298392\pi$
$947$	36.4274	1.18373	0.591866	+	0.806037i	$0.298392\pi$
$948$	0	0
$949$	−36.9339	−1.19893
$950$	0	0
$951$	11.6170	0.376709
$952$	0	0
$953$	34.0870	1.10419	0.552093	−	0.833783i	$-0.313829\pi$
$953$	34.0870	1.10419	0.552093	+	0.833783i	$0.313829\pi$
$954$	0	0
$955$	−13.4046	−0.433761
$956$	0	0
$957$	−4.49966	−0.145453
$958$	0	0
$959$	51.0431	1.64827
$960$	0	0
$961$	−8.78509	−0.283390
$962$	0	0
$963$	−1.83545	−0.0591465
$964$	0	0
$965$	−7.68453	−0.247374
$966$	0	0
$967$	23.5264	0.756558	0.378279	−	0.925692i	$-0.376516\pi$
$967$	23.5264	0.756558	0.378279	+	0.925692i	$0.376516\pi$
$968$	0	0
$969$	−4.87615	−0.156644
$970$	0	0
$971$	−51.5144	−1.65318	−0.826588	−	0.562808i	$-0.809721\pi$
$971$	−51.5144	−1.65318	−0.826588	+	0.562808i	$0.809721\pi$
$972$	0	0
$973$	20.3386	0.652026
$974$	0	0
$975$	−2.77478	−0.0888642
$976$	0	0
$977$	−12.8593	−0.411405	−0.205703	−	0.978615i	$-0.565948\pi$
$977$	−12.8593	−0.411405	−0.205703	+	0.978615i	$0.565948\pi$
$978$	0	0
$979$	2.42249	0.0774230
$980$	0	0
$981$	18.0359	0.575841
$982$	0	0
$983$	−26.3428	−0.840203	−0.420102	−	0.907477i	$-0.638006\pi$
$983$	−26.3428	−0.840203	−0.420102	+	0.907477i	$0.638006\pi$
$984$	0	0
$985$	−1.14448	−0.0364662
$986$	0	0
$987$	−10.6638	−0.339433
$988$	0	0
$989$	66.8257	2.12493
$990$	0	0
$991$	4.04438	0.128474	0.0642370	−	0.997935i	$-0.479539\pi$
$991$	4.04438	0.128474	0.0642370	+	0.997935i	$0.479539\pi$
$992$	0	0
$993$	15.9896	0.507414
$994$	0	0
$995$	−22.9017	−0.726034
$996$	0	0
$997$	−47.2972	−1.49792	−0.748958	−	0.662617i	$-0.769446\pi$
$997$	−47.2972	−1.49792	−0.748958	+	0.662617i	$0.769446\pi$
$998$	0	0
$999$	−3.44575	−0.109019

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2960.2.a.x.1.3		5
4.3	odd	2		1480.2.a.j.1.3	✓	5
20.19	odd	2		7400.2.a.o.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1480.2.a.j.1.3	✓	5	4.3	odd	2
2960.2.a.x.1.3		5	1.1	even	1	trivial
7400.2.a.o.1.3		5	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2960.a (trivial)

\(f(q)\)	\(=\)	\(q-0.612608 q^{3} -1.00000 q^{5} -3.03374 q^{7} -2.62471 q^{9} +O(q^{10})\)	\(q-0.612608 q^{3} -1.00000 q^{5} -3.03374 q^{7} -2.62471 q^{9} +0.741603 q^{11} +4.52946 q^{13} +0.612608 q^{15} +2.10832 q^{17} +3.77534 q^{19} +1.85849 q^{21} -5.48321 q^{23} +1.00000 q^{25} +3.44575 q^{27} +9.90434 q^{29} +4.71327 q^{31} -0.454312 q^{33} +3.03374 q^{35} -1.00000 q^{37} -2.77478 q^{39} -2.99959 q^{41} -12.1873 q^{43} +2.62471 q^{45} -5.73788 q^{47} +2.20358 q^{49} -1.29158 q^{51} +9.09209 q^{53} -0.741603 q^{55} -2.31281 q^{57} -5.06692 q^{59} -15.0296 q^{61} +7.96269 q^{63} -4.52946 q^{65} -3.55869 q^{67} +3.35906 q^{69} -10.1542 q^{71} -8.15417 q^{73} -0.612608 q^{75} -2.24983 q^{77} +2.00913 q^{79} +5.76324 q^{81} -11.3210 q^{83} -2.10832 q^{85} -6.06748 q^{87} +3.26656 q^{89} -13.7412 q^{91} -2.88739 q^{93} -3.77534 q^{95} +2.18735 q^{97} -1.94649 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} - 5 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) \) 5 * q - q^3 - 5 * q^5 + 2 * q^7 + 6 * q^9	\( 5 q - q^{3} - 5 q^{5} + 2 q^{7} + 6 q^{9} - 8 q^{11} - 4 q^{13} + q^{15} - q^{17} - 10 q^{19} + 5 q^{21} - 4 q^{23} + 5 q^{25} - q^{27} + 11 q^{29} + 3 q^{31} - 3 q^{33} - 2 q^{35} - 5 q^{37} - 18 q^{39} + 16 q^{41} - 31 q^{43} - 6 q^{45} - 5 q^{47} + 7 q^{49} - 34 q^{51} + 8 q^{53} + 8 q^{55} - 8 q^{57} - 24 q^{59} - 15 q^{61} - 19 q^{63} + 4 q^{65} - 12 q^{67} + 10 q^{69} - 5 q^{71} + 5 q^{73} - q^{75} - 4 q^{77} - 4 q^{79} + 17 q^{81} - 27 q^{83} + q^{85} + 4 q^{87} + 16 q^{89} - 26 q^{91} + 14 q^{93} + 10 q^{95} - 19 q^{97} - 37 q^{99}+O(q^{100}) \) 5 * q - q^3 - 5 * q^5 + 2 * q^7 + 6 * q^9 - 8 * q^11 - 4 * q^13 + q^15 - q^17 - 10 * q^19 + 5 * q^21 - 4 * q^23 + 5 * q^25 - q^27 + 11 * q^29 + 3 * q^31 - 3 * q^33 - 2 * q^35 - 5 * q^37 - 18 * q^39 + 16 * q^41 - 31 * q^43 - 6 * q^45 - 5 * q^47 + 7 * q^49 - 34 * q^51 + 8 * q^53 + 8 * q^55 - 8 * q^57 - 24 * q^59 - 15 * q^61 - 19 * q^63 + 4 * q^65 - 12 * q^67 + 10 * q^69 - 5 * q^71 + 5 * q^73 - q^75 - 4 * q^77 - 4 * q^79 + 17 * q^81 - 27 * q^83 + q^85 + 4 * q^87 + 16 * q^89 - 26 * q^91 + 14 * q^93 + 10 * q^95 - 19 * q^97 - 37 * q^99

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